The set of scalars λ for which the operator shift T−λI fails to be invertible; it captures the total spectral properties of an operator, generalizing eigenvalues to infinite-dimensional and non-normal contexts.

Principle

For a bounded linear operator on a Banach space the spectrum is a nonempty compact subset of the complex plane; it decomposes (in infinite-dimensional settings) into point (eigenvalues), continuous and residual spectrum and satisfies spectral radius formulas relating to operator norms.

Demonstration

For an n×n matrix the spectrum equals the set of roots of the characteristic polynomial (the eigenvalues). For the unilateral shift on ℓ^2 the spectrum is the closed unit disk even though there are no eigenvalues in the open disk; for compact operators nonzero spectrum points are eigenvalues with finite multiplicity accumulating only at zero.

Misapplication

Treating the spectrum always as the set of eigenvalues (valid in finite dimensions but misleading in infinite-dimensional contexts) or ignoring distinctions between point, continuous and residual spectrum when applying functional calculus or stability arguments.

Consequence

Spectral information governs functional calculus (defining f(T) for holomorphic or continuous f on the spectrum), long-term evolution via the spectrum of generators, and resolvent estimates used in PDEs and numerical analysis; spectral radius bounds growth of iterates.

Reversal

The resolvent set is the complement of the spectrum: scalars where T−λI is invertible and the resolvent operator (T−λI)^{-1} is defined; analytic properties of the resolvent reflect spectral features.

Boundary

Definition depends on the operator class and ambient space: finite-dimensional matrices have purely point spectrum equal to eigenvalues; in Banach or Hilbert spaces additional spectral types appear and spectrum depends on topology and completeness; unbounded operators require domain considerations and lead to spectra extending to continuous spectra from self-adjointness or other properties.

Semantic Tension

Spectrum versus numerical range and versus point spectrum: the spectrum encodes invertibility failure and is stable under algebraic operations, while the numerical range captures quadratic form averages and may be larger or smaller; point spectrum (eigenvalues) is only one component of the full spectrum in infinite dimensions.

Synthesis

The spectrum is the full set of scalars that obstruct invertibility of T−λI; it generalizes eigenvalues to a complete spectral invariant that controls functional calculus, stability and the resolvent behavior of the operator.